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Caring about math of equations and math of solutions

A reader of the Tetragraviton blog named nueww highlighted an interesting footnote on page 126 of Polchinski's memories (arXiv):

Morrison came to UCSB from Duke about ten years ago, with a joint position in math and physics. He plays a unique role in tying these subjects together. He and I have an ongoing friendly dispute about whether I know much math (I claim not). I think that the difference goes back to Susskind’s distinction between the mathematics of the equations and the mathematics of the solutions, where I care only about the former.
David Morrison is a very smart string theorist who was trained as a mathematician. Well, he – and others – weren't just trained as mathematicians. I think that they were born and hardwired to think as mathematicians. The memes in the quote above – invented and promoted by Susskind and Polchinski – seem to crisply demystify the difference between the psychology of a mathematician and the psychology of a theoretical physicist.




First, Joe Polchinski is clearly being way too modest. The amount and depth of fascinating "mathematics of solutions" that he has shown in his papers and pedagogical texts is surely huge and if you impartially measured Polchinski's mathematical IQ – even one specialized for the mathematics of solutions – it would end up in the top 1% or 0.1% of the mankind, to say the least.

But there's still a very true core in Polchinski's words, I think. The difference between his thinking, that of a theoretical physicist, and the thinking of a typical mathematician is significant and it boils down to some very different internal drivers and motivations.




Around page 12 of his memories, Polchinski mentions how much time he spent with chess, became a participant of local tournaments, read theoretical books on chess etc., but he wouldn't surpass the level of a "good recreational player" – he's being modest here as well, I would bet, but again, I am convinced he is not lying here, too – which differs from his "Grandmaster" status as a theoretical physicist.

I actually think that his focus on the "mathematics of the equations" is the same reason that also explains his relative "underperformance in chess". (I write about Polchinski but these qualitative observations hold for me as well – not quantitatively, of course: I am not comparing myself to Polchinski in the absolute sense.)

While mathematicians and theoretical physicists may write somewhat similar papers, do similar things in them, sometimes define objects, sometimes solve problems and equations, they have a very different idea about the "beef of their work":
A theoretical physicist mostly considers a deep problem as a solved one once he finds the full equations that govern the problem – and the problem is therefore reduced to some more or less mechanical operations, at least in principle. The actual solutions may already be left to less profound thinkers or computers: they are a matter of brute force which is not too interesting.

A mathematician imagines that the bulk of the work and depth is this actual later search for the solutions – that what he's doing, wants to be doing, that's what he's good at – and considers the previous search for the right problems and ideas to be either a matter of luck, arbitrary, or something that ordinary people may do, or something that has "obvious" answers.
Whenever you order scientific disciplines on a line, you usually push mathematics to the "more abstract, unpractical" end of the axis than the theoretical physicists. But because of the comparison discussed in this article, mathematicians are actually more practical than the theoretical physicists. They are actually imagining that their skills should be used to solve pre-determined problems. Well, that's why you often hear about applied mathematicians. Theoretical physicists can't really be applied because the words theoretical and applied are antonymous.

Theoretical physicists are really focusing on the search what the right problems should be, on finding the relevant and/or interesting rules of the game. So when I learned the rules of chess, I thought that "most of chess has already been mastered". This is obviously not shared by most people – the fun is only getting started once you learn the rules – but I do think that this attitude of mine is more or less defining for the psyche of a theoretical physicist.

So I think that Joe Polchinski could become a chess grandmaster if the intelligence were the only thing that mattered. But it's some internal difference in motivations that decides otherwise. I think that his subconscious mechanisms tell the rest of his mind that "it's ultimately a waste of mental energy" to do the mechanical operations such as the scanning of the space of possible future moves in chess.

Polchinski himself explains his being "a physics Grandmaster but not a chess Grandmaster" by his less than stellar memory and the fear of irreversible moves. But I think that those aren't the "primary" differences. His memory is being cleaned subconsciously but intentionally and the fear of irreversible moves follows from some kind of theoretical perfectionism which differs from the "let's live" trial-and-error paradigm.

Chess is often counted as a sport, a mind sport of a sort. Many physically oriented people laugh. Chess is a sport? That's funny. When an intellectually oriented young person would choose chess as his sport of choice, they would think it's a swindle. Chess is like some kind of mathematics, isn't it? But at the end, I think that they are wrong. Chess is a sport. One becomes good at it if he deepens his skills that may be classified as a brute force of a sort.

In physical sports, one wants to be strong, fast etc. In chess, one wants to be strong mentally, have a high CPU capacity and memory and some combinations of them. But in both cases, the motivation is sports-like. Well, people like Polchinski or myself don't have enough of this motivation. It looks too egotist, too narrow-minded. I just don't get why I "should" be a better chess player or Olympic sprinter than someone else. What would be better about the world? Needless to say, my chances would be 0 to achieve the Olympic level in physical sports but "somewhat higher yet still low" for mind sports.

But isn't it irrelevant who is the fastest sprinter in the world? He may be faster by one percent than his top competitors. But he may earn 10 times as much for that. Is that fair? Why does someone earn 10 times more for being 1% faster? Needless to say, similar questions may be asked about all other sports – and to a large extent, that includes the mind sports such as chess, too. I am just utterly unimpressed about one man's being 1% faster than another man. Why should it matter? And isn't the careful following of these 1% improvements more boring than the most boring bureaucratic work of a secretary? The whole concept of making someone insanely rich or famous because of these tiny relative differences looks like a sign of the mankind's collective irrationality to me. It doesn't mean that I never watch sports and I never find it fun. It is often fun. But despite the fun, I still rationally realize that this fun is irrational.

Well, I am unimpressed even when one man is 30% faster than another man. They're still comparable. It's still a sign of the mankind's collective irrationality for the first man to earn 1,000 times and sometimes 1,000,000 more in sports than the other one. After all, robots already greatly surpass humans in physical sports as well as mind sports, don't they? So why would humans be so obsessed with some disciplines in which they're not too good even as a species? And think about the sex gap. If sports weren't segregated, women would be earning literally zero as athletes – just because of these 30%-like differences. So the whole income of a rather large group of people – female athletes – depends on a pure sociological convention, the segregation of sexes. Doesn't this dependence on social conventions say something unflattering about the whole idea of professional sports?

What's different about theoretical physics is that the skills, talents, and actual work that top theoretical physicists may display or perform may be greater than other people's skills and work – and not just by 1% or 30%. They may be and they often are larger by many orders of magnitude, sometimes a dozen of orders of magnitude. For all practical and most of the impractical purposes, a man familiar with quantum mechanics belongs to a different species than a man who is stuck in the classical thinking. And quantum mechanics is not the only "gap" that separates the people to these very different "castes".

Breakthroughs in theoretical physics may change and sometimes do change the rules of the game fundamentally. They're not like improving the fastest 100-meter sprint by a fraction of a percent.

And the difference between the two psychologies isn't just about the magnitude of the breakthrough. It's about its "universal relevance". When you beat another sprinter by 0.02 seconds, you're just making a big change from your viewpoint. From a more objective viewpoint, one African sprinter has just trumped another. (I am assuming that this blog post is being read mostly by African sprinters.) What's the difference? ;-) But the advances in theoretical physics are "big changes" even from an objective viewpoint, even if you don't care about the precise names of the people who make them and the differences between these people.

At the end, I think that the excessive modesty of folks like Polchinski is bad news. Folks like Polchinski are making a huge difference but the "majority opinion in the society" understates the importance of their work – and the skills and talents that are needed for that work – dramatically.

No, the right rules of the game – the fundamental equations and rules of physics in particular – aren't obvious to start with. And no, they won't be found by an average person who is just a little bit lucky. The discoveries of such things are transformative events that decide about all the minor ones.

I must mention that when I was a teenager, I was greatly influenced by (the Czech translation of) a letter that Einstein wrote for Max Planck's 60th birthday:
In the temple of science are many mansions, and various indeed are they that dwell therein and the motives that have led them thither. Many take to science out of a joyful sense of superior intellectual power; science is their own special sport to which they look for vivid experience and the satisfaction of ambition; many others are to be found in the temple who have offered the products of their brains on this altar for purely utilitarian purposes. Were an angel of the Lord to come and drive all the people belonging to these two categories out of the temple, the assemblage would be seriously depleted, but there would still be some men, of both present and past times, left inside. Our Planck is one of them, and that is why we love him.

I am quite aware that we have just now light-heartedly expelled in imagination many excellent men who are largely, perhaps chiefly, responsible for the building of the temple of science; and in many cases our angel would find it a pretty ticklish job to decide. But of one thing I feel sure: if the types we have just expelled were the only types there were, the temple would never have come to be, any more than a forest can grow which consists of nothing but creepers. For these people any sphere of human activity will do, if it comes to a point; whether they become engineers, officers, tradesmen, or scientists depends on circumstances. Now let us have another look at those who have found favor with the angel. Most of them are somewhat odd, uncommunicative, solitary fellows, really less like each other, in spite of these common characteristics, than the hosts of the rejected. What has brought them to the temple? That is a difficult question and no single answer will cover it. To begin with, I believe with Schopenhauer that one of the strongest motives that leads men to art and science is escape from everyday life with...
As you can see, the angel expelled all the superficial people, the athletes, chess players, and careerists of all kinds. After this expulsion, folks like Planck, Einstein, and Polchinski remained in that place. In 1918, Einstein found it both safe and natural to talk about the careerists and athletes of science as if they were weeds or creepers. It's too bad that during the following 99 years, the counterparts, followers, and disciples of Einstein were basically turned to modest guys who aren't proud about what they are and/or who have to hide this pride.

Well, I think that e.g. in the case of Joe, it's more about the hiding. Also, it seems to me that folks like Morrison have been fooled by this superficially modest talk and they still haven't gotten a key point – that Polchinski is actually intrinsically proud about "being weaker in the mathematics of solutions" because he ultimately knows it's a positive trait. So, Dave, if you're telling a physicist like Polchinski that he (Polchinski) is good at mathematics in a similar sense as mathematicians (or you), you are not actually flattering him! ;-)

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